I have a 2008 MacBook. 2-3 minutes after startup the rotating beach ball appears. The only option at that point is to reboot. I have tried reinstalling the factory disks, installing Snow Leopard 10.6.3. I have replaced the HDD with a factory Fujitsu. It passes Apple Hardware Test (AHT). I have also created a USB bootable and booted from there. Same results, even with the HDD removed. Beyond that I have been through all the following: https://support.apple.com/en-us/HT201255 (start up options), https://support.apple.com/en-us/HT203161 (troubleshooting). The only mode in which it will operate without crashing is in single user mode. I've been working on it for 3 weeks and I'm afraid I'm going to have to throw it away.
It seems like you've covered almost everything, but what I would do is start with the RAM. You said haven't changed it and RAM problems typically leave the computer somewhat functional, like how it will startups, but soon crash. I can personally recommend OWC for Mac replacement parts, but wherever you go, just make sure to buy Mac-specific RAM; I've had a terrible experience with Best Buy, one-size-fits-all RAM.
If that doesn't do it, it almost has to be the logic board, but don't go there first, as it should cost you more than RAM.